L(s) = 1 | + (0.944 − 0.328i)5-s + (−0.524 + 0.851i)7-s + (−0.561 + 0.827i)11-s + (−0.272 + 0.962i)13-s + (0.642 − 0.766i)17-s + (−0.300 − 0.953i)19-s + (−0.524 − 0.851i)23-s + (0.784 − 0.620i)25-s + (−0.858 + 0.512i)29-s + (−0.286 − 0.957i)31-s + (−0.216 + 0.976i)35-s + (−0.976 + 0.216i)37-s + (0.929 − 0.369i)41-s + (0.997 + 0.0726i)43-s + (−0.957 − 0.286i)47-s + ⋯ |
L(s) = 1 | + (0.944 − 0.328i)5-s + (−0.524 + 0.851i)7-s + (−0.561 + 0.827i)11-s + (−0.272 + 0.962i)13-s + (0.642 − 0.766i)17-s + (−0.300 − 0.953i)19-s + (−0.524 − 0.851i)23-s + (0.784 − 0.620i)25-s + (−0.858 + 0.512i)29-s + (−0.286 − 0.957i)31-s + (−0.216 + 0.976i)35-s + (−0.976 + 0.216i)37-s + (0.929 − 0.369i)41-s + (0.997 + 0.0726i)43-s + (−0.957 − 0.286i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.501 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5184 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.501 - 0.865i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3326921071 - 0.5770469487i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3326921071 - 0.5770469487i\) |
\(L(1)\) |
\(\approx\) |
\(0.9588217976 + 0.004240827159i\) |
\(L(1)\) |
\(\approx\) |
\(0.9588217976 + 0.004240827159i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (0.944 - 0.328i)T \) |
| 7 | \( 1 + (-0.524 + 0.851i)T \) |
| 11 | \( 1 + (-0.561 + 0.827i)T \) |
| 13 | \( 1 + (-0.272 + 0.962i)T \) |
| 17 | \( 1 + (0.642 - 0.766i)T \) |
| 19 | \( 1 + (-0.300 - 0.953i)T \) |
| 23 | \( 1 + (-0.524 - 0.851i)T \) |
| 29 | \( 1 + (-0.858 + 0.512i)T \) |
| 31 | \( 1 + (-0.286 - 0.957i)T \) |
| 37 | \( 1 + (-0.976 + 0.216i)T \) |
| 41 | \( 1 + (0.929 - 0.369i)T \) |
| 43 | \( 1 + (0.997 + 0.0726i)T \) |
| 47 | \( 1 + (-0.957 - 0.286i)T \) |
| 53 | \( 1 + (0.130 - 0.991i)T \) |
| 59 | \( 1 + (0.187 - 0.982i)T \) |
| 61 | \( 1 + (0.355 + 0.934i)T \) |
| 67 | \( 1 + (0.244 + 0.969i)T \) |
| 71 | \( 1 + (-0.422 + 0.906i)T \) |
| 73 | \( 1 + (-0.422 - 0.906i)T \) |
| 79 | \( 1 + (-0.918 + 0.396i)T \) |
| 83 | \( 1 + (-0.999 + 0.0145i)T \) |
| 89 | \( 1 + (-0.906 + 0.422i)T \) |
| 97 | \( 1 + (0.0581 + 0.998i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−18.145493633008897387820018094235, −17.45464907662023010715314186663, −16.98423406581783261998286060568, −16.29865770165753913028439600279, −15.63738540913136058728718716247, −14.74461565296906685045815640598, −14.11368053967096788142451096312, −13.645689206410088200143132863035, −12.7893932277691347029550226896, −12.581333353608841688526593098919, −11.297423688022533530920852090805, −10.59262739912178534073715416164, −10.2295497851379162313734596951, −9.65507552629753553073798407389, −8.75771768679173638104470968990, −7.8384962297118947131923335735, −7.44268950827094400244816077203, −6.41436963376174556038539007839, −5.74571863189504316509872995378, −5.46484585349128661495721239946, −4.19891272243554931472511317937, −3.39108368375825887336103864445, −2.94056457251135214037999825747, −1.838320253492528598583421629400, −1.10302668033534822261529070517,
0.16515628445850851267812977054, 1.52354599408302552601283181001, 2.400239426461873857587128380313, 2.59266487369099671099614327908, 3.88939076922945924173622437291, 4.78468038564895234342199996905, 5.32902316672009048502767018984, 6.01133795931704042225084288895, 6.82654093504302282898606677764, 7.34961810445016215686274224395, 8.50418208183062476501020894676, 9.04412622260085138566796944066, 9.72710308205625414411728633572, 10.03535280098527265195779208901, 11.11047683425849219623518674402, 11.82449210759365360552033539007, 12.6454809551724300558688703230, 12.905827580892854820610965080459, 13.75600093977061495453135406571, 14.49961210900951168063365984557, 14.98699759454496373548722686232, 16.06612493426176190411014903520, 16.23714085805132415271357484452, 17.18596777131504754727429919371, 17.758320212536293113895562869950